Problem: What is the remainder when $11065+11067+11069+11071+11073+11075+11077$ is divided by $14$?
Explanation: Since $11065,11067,11069,\ldots,11077$ are $7$ consecutive odd integers, they include exactly one integer from each of the residue classes $1,3,5,7,9,11,13\pmod{14}$ (not necessarily in that order). Therefore, their sum is congruent $\pmod{14}$ to $1+3+5+7+9+11+13=49$. The remainder of this sum $\pmod{14}$ is $\boxed{7}$.